Absolute values and tensor powers of irreducible characters

Let $\chi$ be a character of a complex irreducible representation of a finite group $G$. We present a simple formula for the expectation of the random variable $(\mathrm{\mid }\chi \mathrm{\mid }\mathrm{/}\chi (1){)}^t$ in terms of character ratios $(\mathrm{\mid }\chi (g)\mathrm{\mid }\mathrm{/}\chi (1){)}^t,g\in G,t\ge 0$. As a follow up we briefly discuss asymptotic properties of the formula and its relation to the subject of growth of dimensions of isotypic components in (virtual) tensor powers of irreducible representations. Similar type of reasoning can be applied to some questions related to commuting probability. In particular, we obtain an analogue of Frobenis formula for the probability of an “event” $\mathrm{\mid }\chi ([x,y{]}^{-1}g)\mathrm{\mid }=r$